Timeline for Zariski tangent space of a scheme as the vector space of derivations
Current License: CC BY-SA 3.0
7 events
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| Feb 7, 2012 at 14:21 | comment | added | user2035 | Derivations are also nontrivial for non-closed points of varieties. The general case, say $f\colon X\to S$ and $f(x)=s$, may be reduced to the field case by considering the relative Zariski tangent space $\mathfrak m_x/(\mathfrak m_x^2+\mathfrak m_s\mathcal O_{X,x})$ instead and noticing $\Omega_{X_s,x/\kappa(s)}=\Omega_{X,x}\otimes\kappa(s)$. | |
| Feb 2, 2012 at 17:32 | comment | added | Dima Sustretov | @David Speyer: Thank you for the example. I was rather wondering if the failure of this lemma is described by some theory, similarly to the Jacobson's "Galois" theory for fields lying between $k$ and $k^p$ for imperfect fields. Looking at the conormal sequence helps in the ground field case, as a-fortiori pointed out. I wonder if something can be said about some class of schemes not over a field too. | |
| Feb 2, 2012 at 17:20 | comment | added | Dima Sustretov | @a-fortiory: I see, there is no reason for it to be injective in general. It follows from the conormal sequnce for $\mathcal{O}_{X,x}$, $\mathfrak m$ and $k$ that it is non-injective if $\Omega(\mathcal{O}_{X,x}/\mathfrak{m},k)$ is non-zero, which you can only expect in a non-separable case. | |
| Feb 1, 2012 at 13:39 | comment | added | David E Speyer | You can take $X = \mathrm{Spec} \mathbb{Z}_p$ and $\mathfrak{m} = (p)$. Then $\mathrm{Der}(\mathbb{Z}_p, \mathbb{F}_p)=0$. Is this the sort of thing you are looking for? | |
| Feb 1, 2012 at 11:17 | history | edited | Dima Sustretov | CC BY-SA 3.0 |
deleted 29 characters in body; deleted 17 characters in body
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| Jan 31, 2012 at 19:40 | comment | added | user2035 | Actually, the homomorphism $\mathrm{Der}(\mathcal O_{X,x},\kappa(x))\to(\mathfrak m/\mathfrak m^2)^*$ is not always injective. For a characterization of surjectivity in the case of a ground field, see Eisenbud, Commutative Algebra, Corollary 16.13. | |
| Jan 31, 2012 at 17:41 | history | asked | Dima Sustretov | CC BY-SA 3.0 |